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%I M0043 N0013 #73 Oct 11 2022 07:41:15
%S 1,1,2,1,0,2,0,1,3,0,2,2,0,0,0,1,2,3,2,0,0,2,0,2,1,0,4,0,0,0,0,1,4,2,
%T 0,3,0,2,0,0,2,0,2,2,0,0,0,2,1,1,4,0,0,4,0,0,4,0,2,0,0,0,0,1,0,4,2,2,
%U 0,0,0,3,2,0,2,2,0,0,0,0,5,2,2,0,0,2,0,2,2,0,0,0,0,0,0,2,2,1,6,1,0,4,0,0,0
%N Glaisher's J numbers.
%C Number of integer solutions to the equation x^2 + 2*y^2 = n when (-x, -y) and (x, y) are counted as the same solution.
%C For n nonzero, a(n) is nonzero if and only if n is in A002479. - _Michael Somos_, Dec 15 2011
%C Coefficients of Dedekind zeta function for the quadratic number field of discriminant -8. See A002324 for formula and Maple code. - _N. J. A. Sloane_, Mar 22 2022
%D B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 114 Entry 8(iii).
%D L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 19.
%D N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.24).
%D J. W. L. Glaisher, Table of the excess of the number of (8k+1)- and (8k+3)-divisors of a number over the number of (8k+5)- and (8k+7)-divisors, Messenger Math., 31 (1901), 82-91.
%D D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002325/b002325.txt">Table of n, a(n) for n = 1..10000</a>
%H J. W. L. Glaisher, <a href="/A002325/a002325.pdf">Table of the excess of the number of (8k+1)- and (8k+3)-divisors of a number over the number of (8k+5)- and (8k+7)-divisors</a>, Messenger Math., 31 (1901), 82-91. [Incomplete annotated scanned copy]
%H Michael D. Hirschhorn, <a href="http://dx.doi.org/10.1016/j.disc.2004.08.045">The number of representations of a number by various forms</a>, Discrete Mathematics 298 (2005), 205-211.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%H <a href="/index/Ge#Glaisher">Index entries for sequences related to Glaisher's numbers</a>.
%F Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m, p)+1)*p^(-s) + Kronecker(m, p)*p^(-2s))^(-1) for m = -2.
%F Moebius transform is period 8 sequence [ 1, 0, 1, 0, -1, 0, -1, 0, ...]. - _Michael Somos_, Aug 23 2005
%F G.f.: (theta_3(q) * theta_3(q^2) - 1) / 2 = Sum_{k>0} Kronecker( -2, n) * x^k / (1 - x^k) = Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(4*k)).
%F Multiplicative with a(2^e) = 1, a(p^e) = e+1 if p == 1, 3 (mod 8), a(p^e) = (1+(-1)^e)/2 if p == 5, 7 (mod 8). - _Michael Somos_, Oct 23 2006
%F A033715(n) = 2 * a(n) unless n=0.
%F a(n) = A188169(n) + A188170(n) - A188171(n) - A188172(n) [Hirschhorn]. - _R. J. Mathar_, Mar 23 2011
%F G.f.: A(x) = 2*(1+x^2)/(G(0)-2*x*(1+x^2)); G(k) = 1+x+x^(2*k)*(1+x^3+x^(2*k+1)+x^(2*k+4)+x^(4*k+3)+x^(4*k+4)) - x*(1+x^(2*k))*(1+x^(2*k+4))*(1+x^(4*k+4))^2/G(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jan 03 2012
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(2)) = 1.110720... (A093954). - _Amiram Eldar_, Oct 11 2022
%e x + x^2 + 2*x^3 + x^4 + 2*x^6 + x^8 + 3*x^9 + 2*x^11 + 2*x^12 + x^16 + ...
%p S:= series( (JacobiTheta3(0,q)*JacobiTheta3(0,q^2)-1)/2, q, 1001):
%p seq(coeff(S,q,j), j=1..1000); # _Robert Israel_, Dec 01 2015
%t a[n_] := Total[ KroneckerSymbol[-8, #] & /@ Divisors[n]]; Table[a[n], {n, 1, 105}] (* _Jean-François Alcover_, Nov 25 2011, after _Michael Somos_ *)
%t QP = QPochhammer; s = ((QP[q^2]^3*QP[q^4]^3)/(QP[q]^2*QP[q^8]^2)-1)/(2q) + O[q]^105; CoefficientList[s, q] (* _Jean-François Alcover_, Dec 01 2015, adapted from PARI *)
%o (PARI) a(n) = if( n<1, 0, issquare(n)-issquare(2*n) + 2*sum(i=1,sqrtint(n\2), issquare(n-2*i^2)))
%o (PARI) {a(n) = if( n<1, 0, qfrep([ 1, 0; 0, 2],n)[n])} \\ _Michael Somos_, Jun 05 2005
%o (PARI) {a(n) = if( n<1, 0, direuler(p=2, n, 1 / (1 - X) / (1 - kronecker( -2, p) * X))[n])} \\ _Michael Somos_, Jun 05 2005
%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -2, d)))} \\ _Michael Somos_, Aug 23 2005
%o (PARI) {a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 1, if( p%8<4, e+1, !(e%2))))))} \\ _Michael Somos_, Oct 23 2006
%o (PARI) {a(n) = local(A); if( n<1, 0, A = x * O(x^n); polcoeff( eta(x + A)^-2 * eta(x^2 + A)^3 * eta(x^4 + A)^3 * eta(x^8 + A)^-2, n) / 2)}
%o (PARI) a(n) = my(f=factor(n>>valuation(n,2)), e); prod(i=1, #f~, e=f[i, 2]; if( f[i, 1]%8<4, e+1, 1 - e%2)) \\ _Charles R Greathouse IV_, Sep 09 2014
%Y Cf. A033715, A093954.
%Y Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
%Y Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.
%K nonn,easy,nice,mult
%O 1,3
%A _N. J. A. Sloane_
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